double LeastMedianOfSquares(const Kernel &kernel,
typename Kernel::Model * model = NULL,
double* outlierThreshold = NULL,
double outlierRatio=0.5,
double minProba=0.99)
{
const size_t min_samples = Kernel::MINIMUM_SAMPLES;
const size_t total_samples = kernel.NumSamples();
std::vector<double> residuals(total_samples); // Array for storing residuals
std::vector<size_t> vec_sample(min_samples);
double dBestMedian = std::numeric_limits<double>::max();
// Required number of iterations is evaluated from outliers ratio
const size_t N = (min_samples<total_samples)?
getNumSamples(minProba, outlierRatio, min_samples): 0;
for (size_t i=0; i < N; i++) {
// Get Samples indexes
UniformSample(min_samples, total_samples, &vec_sample);
// Estimate parameters: the solutions are stored in a vector
std::vector<typename Kernel::Model> models;
kernel.Fit(vec_sample, &models);
// Now test the solutions on the whole data
for (size_t k = 0; k < models.size(); ++k) {
//Compute Residuals :
for (size_t l = 0; l < total_samples; ++l) {
double error = kernel.Error(l, models[k]);
residuals[l] = error;
}
// Compute median
std::vector<double>::iterator itMedian = residuals.begin() +
std::size_t( total_samples*(1.-outlierRatio) );
std::nth_element(residuals.begin(), itMedian, residuals.end());
double median = *itMedian;
// Store best solution
if(median < dBestMedian) {
dBestMedian = median;
if (model) (*model) = models[k];
}
}
}
// This array of precomputed values corresponds to the inverse
// cumulative function for a normal distribution. For more information
// consult the litterature (Robust Regression for Outlier Detection,
// rouseeuw-leroy). The values are computed for each 5%
static const double ICDF[21] = {
1.4e16, 15.94723940, 7.957896558, 5.287692054,
3.947153876, 3.138344200, 2.595242369, 2.203797543,
1.906939402, 1.672911853, 1.482602218, 1.323775627,
1.188182950, 1.069988721, 0.9648473415, 0.8693011162,
0.7803041458, 0.6946704675, 0.6079568319,0.5102134568,
0.3236002672
};
// Evaluate the outlier threshold
if(outlierThreshold) {
double sigma = ICDF[int((1.-outlierRatio)*20.)] *
(1. + 5. / double(total_samples - min_samples));
*outlierThreshold = (double)(sigma * sigma * dBestMedian * 4.);
if (N==0) *outlierThreshold = std::numeric_limits<double>::max();
}
return dBestMedian;
}
/// Generic implementation of 'ORSA':
/// A Probabilistic Criterion to Detect Rigid Point Matches
/// Between Two Images and Estimate the Fundamental Matrix.
/// Bibtex :
/// @article{DBLP:journals/ijcv/MoisanS04,
/// author = {Lionel Moisan and B{\'e}renger Stival},
/// title = {A Probabilistic Criterion to Detect Rigid Point Matches
/// Between Two Images and Estimate the Fundamental Matrix},
/// journal = {International Journal of Computer Vision},
/// volume = {57},
/// number = {3},
/// year = {2004},
/// pages = {201-218},
/// ee = {http://dx.doi.org/10.1023/B:VISI.0000013094.38752.54},
/// bibsource = {DBLP, http://dblp.uni-trier.de}
///}
///
/// ORSA is based on an a contrario criterion of
/// inlier/outlier discrimination, is parameter free and relies on an optimized
/// random sampling procedure. It returns the log of NFA and optionally
/// the best estimated model.
///
/// \param vec_inliers Output vector of inlier indices.
/// \param nIter The number of iterations.
/// \param precision (input/output) threshold for inlier discrimination.
/// \param model The best computed model.
/// \param bVerbose Display optimization statistics.
double OrsaModel::orsa(std::vector<int> & vec_inliers,
size_t nIter,
double *precision,
Model *model,
bool bVerbose) const {
vec_inliers.clear();
const int sizeSample = SizeSample();
const int nData = x1_.ncol();
if(nData <= sizeSample)
return std::numeric_limits<double>::infinity();
const double maxThreshold = (precision && *precision>0)?
*precision * *precision *N2_(0,0)*N2_(0,0): // Square max error
std::numeric_limits<double>::infinity();
std::vector<ErrorIndex> vec_residuals(nData); // [residual,index]
std::vector<int> vec_sample(sizeSample); // Sample indices
// Possible sampling indices (could change in the optimization phase)
std::vector<int> vec_index(nData);
for (int i = 0; i < nData; ++i)
vec_index[i] = i;
// Precompute log combi
double loge0 = log10((double)NbModels() * (nData-sizeSample));
std::vector<float> vec_logc_n, vec_logc_k;
makelogcombi_n(nData, vec_logc_n);
makelogcombi_k(sizeSample,nData, vec_logc_k);
// Reserve 10% of iterations for focused sampling
size_t nIterReserve=nIter/10;
nIter -= nIterReserve;
// Output parameters
double minNFA = std::numeric_limits<double>::infinity();
double errorMax = 0;
int side=0;
// Main estimation loop.
for (size_t iter=0; iter < nIter; iter++) {
UniformSample(sizeSample, vec_index, &vec_sample); // Get random sample
std::vector<Model> vec_models; // Up to max_models solutions
Fit(vec_sample, &vec_models);
// Evaluate models
bool better=false;
for (size_t k = 0; k < vec_models.size(); ++k)
{
// Residuals computation and ordering
for (int i = 0; i < nData; ++i)
{
int s;
double error = Error(vec_models[k], i, &s);
vec_residuals[i] = ErrorIndex(error, i, s);
}
std::sort(vec_residuals.begin(), vec_residuals.end());
// Most meaningful discrimination inliers/outliers
ErrorIndex best = bestNFA(vec_residuals, loge0, maxThreshold,
vec_logc_n, vec_logc_k);
if(best.error < minNFA) // A better model was found
{
better = true;
minNFA = best.error;
side = best.side;
vec_inliers.resize(best.index);
for (int i=0; i<best.index; ++i)
vec_inliers[i] = vec_residuals[i].index;
errorMax = vec_residuals[best.index-1].error; // Error threshold
if(best.error<0 && model) *model = vec_models[k];
if(bVerbose)
{
std::cout << " nfa=" << minNFA
<< " inliers=" << vec_inliers.size()
<< " precision=" << denormalizeError(errorMax, side)
<< " im" << side+1
<< " (iter=" << iter;
if(best.error<0) {
std::cout << ",sample=" << vec_sample.front();
std::vector<int>::const_iterator it=vec_sample.begin();
for(++it; it != vec_sample.end(); ++it)
std::cout << ',' << *it;
}
std::cout << ")" <<std::endl;
}
}
}
// ORSA optimization: draw samples among best set of inliers so far
if((better && minNFA<0) || (iter+1==nIter && nIterReserve)) {
if(vec_inliers.empty()) { // No model found at all so far
nIter++; // Continue to look for any model, even not meaningful
nIterReserve--;
} else {
vec_index = vec_inliers;
if(nIterReserve) {
nIter = iter+1+nIterReserve;
nIterReserve=0;
}
}
}
}
if(minNFA >= 0)
vec_inliers.clear();
if(bConvergence)
refineUntilConvergence(vec_logc_n, vec_logc_k, loge0,
maxThreshold, minNFA, model, bVerbose, vec_inliers,
errorMax, side);
if(precision)
*precision = denormalizeError(errorMax, side);
if(model && !vec_inliers.empty())
Unnormalize(model);
return minNFA;
}
std::pair<double, double> ACRANSAC(const Kernel &kernel,
std::vector<size_t> & vec_inliers,
size_t nIter = 1024,
typename Kernel::Model * model = NULL,
double precision = std::numeric_limits<double>::infinity(),
bool bVerbose = false)
{
vec_inliers.clear();
const size_t sizeSample = Kernel::MINIMUM_SAMPLES;
const size_t nData = kernel.NumSamples();
if(nData <= (size_t)sizeSample)
return std::make_pair(0.0,0.0);
const double maxThreshold = (precision==std::numeric_limits<double>::infinity()) ?
std::numeric_limits<double>::infinity() :
precision * kernel.normalizer2()(0,0) * kernel.normalizer2()(0,0);
std::vector<ErrorIndex> vec_residuals(nData); // [residual,index]
std::vector<double> vec_residuals_(nData);
std::vector<size_t> vec_sample(sizeSample); // Sample indices
// Possible sampling indices (could change in the optimization phase)
std::vector<size_t> vec_index(nData);
for (size_t i = 0; i < nData; ++i)
vec_index[i] = i;
// Precompute log combi
double loge0 = log10((double)Kernel::MAX_MODELS * (nData-sizeSample));
std::vector<float> vec_logc_n, vec_logc_k;
makelogcombi_n(nData, vec_logc_n);
makelogcombi_k(sizeSample, nData, vec_logc_k);
// Output parameters
double minNFA = std::numeric_limits<double>::infinity();
double errorMax = std::numeric_limits<double>::infinity();
// Reserve 10% of iterations for focused sampling
size_t nIterReserve = nIter/10;
nIter -= nIterReserve;
// Main estimation loop.
for (size_t iter=0; iter < nIter; ++iter) {
UniformSample(sizeSample, vec_index, &vec_sample); // Get random sample
std::vector<typename Kernel::Model> vec_models; // Up to max_models solutions
kernel.Fit(vec_sample, &vec_models);
// Evaluate models
bool better = false;
for (size_t k = 0; k < vec_models.size(); ++k) {
// Residuals computation and ordering
kernel.Errors(vec_models[k], vec_residuals_);
for (size_t i = 0; i < nData; ++i) {
const double error = vec_residuals_[i];
vec_residuals[i] = ErrorIndex(error, i);
}
std::sort(vec_residuals.begin(), vec_residuals.end());
// Most meaningful discrimination inliers/outliers
const ErrorIndex best = bestNFA(
sizeSample,
kernel.logalpha0(),
vec_residuals,
loge0,
maxThreshold,
vec_logc_n,
vec_logc_k,
kernel.multError());
if (best.first < minNFA /*&& vec_residuals[best.second-1].first < errorMax*/) {
// A better model was found
better = true;
minNFA = best.first;
vec_inliers.resize(best.second);
for (size_t i=0; i<best.second; ++i)
vec_inliers[i] = vec_residuals[i].second;
errorMax = vec_residuals[best.second-1].first; // Error threshold
if(model) *model = vec_models[k];
if(bVerbose) {
std::cout << " nfa=" << minNFA
<< " inliers=" << best.second
<< " precisionNormalized=" << errorMax
<< " precision=" << kernel.unormalizeError(errorMax)
<< " (iter=" << iter;
std::cout << ",sample=";
std::copy(vec_sample.begin(), vec_sample.end(),
std::ostream_iterator<size_t>(std::cout, ","));
std::cout << ")" <<std::endl;
}
}
}
// ACRANSAC optimization: draw samples among best set of inliers so far
if((better && minNFA<0) || (iter+1==nIter && nIterReserve)) {
if(vec_inliers.empty()) { // No model found at all so far
nIter++; // Continue to look for any model, even not meaningful
nIterReserve--;
} else {
// ACRANSAC optimization: draw samples among best set of inliers so far
vec_index = vec_inliers;
if(nIterReserve) {
nIter = iter+1+nIterReserve;
nIterReserve=0;
}
}
}
}
if(minNFA >= 0)
vec_inliers.clear();
if (!vec_inliers.empty())
{
if (model)
kernel.Unnormalize(model);
errorMax = kernel.unormalizeError(errorMax);
}
return std::make_pair(errorMax, minNFA);
}
